σ-Fields

Defining a probability space

Definition: Probability space

A probability space is a triplet $(\Omega, \cF, \Pr)$, where $\Omega$ is called the sample space, $\cF$ is called a $\sigma$-Field and $\Pr$ is called the probability measure.

Sample space

This is the set of all possible outcomes of a random experiment. For example, consider rolling a $6$-sided die. Then the sample space is $\Omega = \{1, 2, 3, 4, 5, 6\}$.

Defining a σ-field

Definition: σ-field

A set of subsets of $\Omega$, denoted by $\cF$, is called a $\sigma$-field on $\Omega$ if it satisfies the following properties:

• $\cF$ contains the sample space. $\Omega \in \cF$.
• $\cF$ is closed under complements. $A\in\cF \implies A^c\in \cF$.
• $\cF$ is closed under countable unions. $A_1, A_2, \ldots \in \cF \implies \bigcup_{i=1}^{\infty}A_i\in \cF$.

Examples

The intuitive way to think about a $\sigma$-field is to view it as a collection (set) of events. Every member of $\cF$ is an event. A few examples:

Examples: σ-fields

Example 1

The trivial $\sigma$-Field given by $\cF = \{\emptyset, \Omega\}$, where $\emptyset$ denotes the empty set.

Example 2

Power set of $\Omega$ given by $\cF = \{2^{\Omega}\}$, which denotes the collection of all subsets of $\Omega$.

Example 3

For any fixed subset $A\subset\Omega$, the set given by $\cF = \{\emptyset,A,A^c,\Omega\}$.

Example 4

The set $\cF = \{A: A$ is countable or $A^c$ is countable $\}$.

The four examples above can be verified to be a $\sigma$-field by simply verifying the three properties described above. For example 4, the third property is non-trivial, and can be verified using the following argument:

Consider a countable sequence of events $\{A_i\}_{i=1}^{\infty}\in\cF$. If all $A_i$ are countable, then $B=\cup_i A_i$ is countable and so $B\in\cF$; whereas, if there exists some $j$ such that $A_j$ is uncountable, it means that $A_j^c$ is countable by definition of $\cF$. Now see that since $A_j\subseteq B=\cup_i A_i$, we have $B^c\subseteq A_j^c$. Hence, $B^c$ is countable, implying that $B\in \cF$ by definition.

Counterexample

It is natural at this point to ask for an example of a set which is not a $\sigma$-field. Let’s construct one!

Let $\Omega = (0, 1]$, and define

$$\cB_0 = \{\text{finite unions of disjoint subintervals of }\Omega\}.$$

At first glance, one might believe that $\cB_0$ is a $\sigma$-field, but it is not. It is actually a field (more on that later), but not a $\sigma$-field.

Exercise 1

Show that $\cB_0$ is not a $\sigma$-field by identifying which property (among the three listed in the definition) breaks.